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SCIENTIFICCALCULATOR

TEACHERSGUIDEJULY 1999EL-531RH

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Contents

Generating Sequencesp.39

Train Journeysp.41

Simulated Dicep.43

Mean Dice Scoresp.45

Again and Againp.47

Fibonaccip.49

Factorizing Quadraticsp.51

Triplesp.53

Teachers Guide Part 1 has already been completed. This guide presents Part 2 beginning on page39 (marked with ). We encourage you to put this guide to good use.

Introductionp.2

How to Operatep.3

Number of Bowlingp.4

Down to Onep.6

Reverse the Orderp.8

Different Productsp.10

Sums and Productsp.12

Target 100p.14

Ordering Fractionsp.16

Addting Fractionsp.18

Halfway Betweenp.20

Near Integersp.22

Reshaping Cuboidsp.24

Function Tablesp.26Palindromesp.28

Trial and Improvementp.30

Last Digitsp.32

A Question and Interestp.34

Getting Evenp.37

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2

The use of calculators as a classroom teaching tool is becoming more and more

popular. Contrary to the belief that their use encourages dependency and inhibits

the development of mental skills, research has proven that calculators are highly

unlikely to harm achievement in mathematics and using them can actually improve

the students performance and attitude.* Calculators allow students to quickly gen-erate large amounts of data from which patterns can be spotted, and predictions can

be made and tested. This is an important aspect of the development of mental meth-

ods of calculation. Therefore, priority must be given to create new ways to exploit

the potential of the calculator as an effective learning tool in the classroom.

This Teachers Guide presents several classroom activities that make use of Sharp

scientific calculators. The purpose of these activities is not to introduce the calcula-

tor as a device to relieve the burden of performing difficult calculations, but rather

to develop the students understanding of mathematical concepts and explore areas

of mathematics that would otherwise be inaccessible. Mental methods should al-

ways be considered as a first resort when tackling calculations introduced in these

activities. The development of trial and improvement methods are supported by the

activities as well. We hope you will find them interesting and useful for reinforcing

your students understanding of mathematical concepts.

* Mike Askew & Dylan Williams (1995) Recent Research in Mathematics Education HMSO

Introduction

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How to Operate

2nd function key

Pressing this key will enable the functions writtenin yellow above the calculator buttons.

ON/C, OFF key

Direct function

Mode key

This calculator can operate in three differentmodes as follows.

Written in yellowabove the ON/C key

1. KEY LAYOUT

Mode = 0; normal mode forperforming normal arithmeticand function calculations.

Mode = 1; STAT-1 mode forperforming 1-variablestatistical calculations.

Mode = 2; STAT-2 mode for

performing 2-variablestatistical calculations.

If the calculator fails to operate normally, press the reset

switch on the back to reinitialise the unit. The display formatand calculation mode will return to their initial settings.

RESET

2. RESET SWITCH

Reset switchRESET

2nd function

[Normal mode]

[STAT-1 mode]

[STAT-2 mode]

NOTE:Pressing the reset switch will erase any data stored in memory.

Read Before Using

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Number BowlingJunior high school

Using the calculator Calculator functions used: Subtraction, addition, last answer memory

Objective Read whole numbers and understand that the position of a digit signifies its value.Understand and use the concept of place value in whole numbers.

Explanation of the activity

Think of a 3-digit number and enter it into your calculator.Pretend each digit is a bowling pin.Knock down each pin one at a time, so that your calculator display shows 0.

A: Using subtraction

B: Using addition

(1) Enter a 3-digit number.

(2) Knock down one digit, or pin; i.e. change the lastdigit to a 0.

(3) Knock down the next pin; i.e. change the tens columndigit to 0.

(4) Knock down the pin of the hundreds column.

A: Using subtraction

638

8

30

600

Press the following buttons and then start operation.

6 3 8 =DEG

A N S - 8 =DEG

A N S - 6 0 0 =DEG

A N S - 3 0 =DEG

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Junior high school

(1) Enter a 3-digit number.

(2) Knock down one digit, or pin; i.e. change the last digitto a 0, except this time, do so by adding a number tothe last digit to make it 0.

(3) Knock down the next pin; i.e. change the tens columndigit to 0.

(4) Knock down the pin of the hundreds column.

B: Using addition

Number Bowling

Using the activity in the classroom This activity is a good game for students to play in pairs.One student enters a number in the calculator, and the other student has to knock each digit, orpin, down.

Example:

638 - 8 = 630630 - 30 = 600600 - 600 = 0

Points for students to discuss It is important for students to talk about what they are doing and use the appropriate language, forexample: six hundred and thirty, minus thirty, equals six hundred. Students should be challengedto vary the starting point; i.e. sometimes starting with the hundreds digit and sometimes with thetens digit.

Fur ther Ideas

Play the game using 2-, 4-, or 5-digit numbers according to the ability of the students.

2

60

300


6386 3 8 =

DEG

A N S + 2 =DEG

A N S + 6 0 =DEG

A N S + 3 0 0 =DEG

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Down to OneJunior high school

Using the calculator Calculator functions used: Subtraction, division, last answer memory

Objective Develop a variety of mental methods of computation.

Develop the use of the four operations to solve problems.Use sequence methods of computation when appropriate to a problem.Estimate and approximate solutions to problems.

Explanation of the activity Use the calculator to generate a 3-digit random number.The aim is to get the calculator to display the number1.Players can use any of the numbers1 9 together with any of the keys below:

, , , , , ,

You cannot put numbers together to make 2- or 3-digit numbers.You can use each number only once.The first player to get his/her calculator display to show1 scores five points.If after an agreed time limit no player has reached1, the player who is closest scores two points.

While working on this activity, students should develop their skills of mental mathematics and theirfluency with numerical calculations.

Suppose the random number you generate is 567.

Example A:

The answer is1 and the game is finished.

567 9

7

8


5 6 79 =DEG

A NS7 =DEG

A N S - 8 =DEG

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Junior high schoolDown to One

Using the activity in the classroom Students should be encouraged to estimate the results of calculations and think about the appro-priate operations and numbers to use during the game. Lets start with 864, for example. Thisnumber is divisible by 9, 6, 3 and 2. The equation 846 9 could therefore be a possible first step.This will prompt students to test the divisibility of numbers. Students should also be encouraged tothink about the various strategies they use.

The game could be played between small group of students.

Points for students to discuss For some students, it may be more appropriate to start with a 2-digit number. In this case, the calcula-tor should be set to fixed decimal place mode by pressing the [2ndF] key once and then pressing the[ . ] key, which has FSE written in yellow above it, until FIX is displayed at the top of the calculatorscreen. And press [2ndF] [TAB] and [0] keys. Doing this will round answers to 0 decimal places. Thestarting number can then be generated by multiplying a random number by100.

Fur ther Ideas

Play the game using decimal starting numbers.Give the students a shuffled set of cards numbered from1 to 9 and a shuffled set ofcards numbered10, 20, 30, 40, 50. Students choose five cards from the first set, andtwo cards from the second set. The calculator is then used to generate a random threedigit integer, and the students have to make this total by using the numbers on the cards.

Example B:

You want to subtract 8 from 9, but you cannot since you have already used 8 once.So...

The calculator displays1 and the game is finished.

9

8

3

2

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Reverse the OrderJunior high school

Objective Develop a variety of mental methods of computation.

Estimate and approximate solutions to problems.

Explanation of the activity Enter any 2-digit number into the calculator.Reverse the order of the digits through simple calculator operations.

While working on this activity, students should develop their skills of mental mathematics.They should also be interpreting and generalizing their answers.

Example A:To reverse the order of 58:

Solution: Add 27 to 58 to get 85.

Now try using a 3-digit number.

Example B:Enter 432 into the calculator

Solution: Add 198 to 234 to get 432.

58 27

432 234

234 198


85 58

Using the calculator Calculator functions used: Addition, subtraction

58+ 27=DEG

2 3 4 + 19 8 =DEG

4 3 2 - 2 3 4 =DEG

8 5 - 5 8 =DEG

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Junior high schoolReverse the Order

Using the activity in the classroom This activity is probably best introduced orally to a group of students. Ask the students to enter

any two digit number into their calculators. Then, ask them to find a simple way to reverse theorder of the digits of these numbers. Students may do this by using inverse operations.

Points for students to discuss After trying an example, the students can talk about the operations and numbers that they used.This discussion should lead to the generalization that one way to reverse the order of the digits isto add or subtract a multiple of 9. More able students could be asked to try and prove this gener-alization:

(10a+b) + N = (10b+a)

N = (10b+a) - (10a+b)

N = 9b- 9a= 9(b- a)

Fur ther Ideas

Try using the activity with 3-digit numbers, 4-digit numbers, etc.Choose any 2-digit number, reverse it, and then add the reversed number to the original.What happens? Try this with 3-digit numbers or 4-digit numbers, etc.

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Different ProductsJunior high school

Objective Estimate and approximate solutions to problems.

Explanation of the activity Have the class make up multiplication problems using the digits1, 2, 3 and 4. Each digit can only beused once. Find out what the largest product among the possible answers will be.

While working on this activity, students should practice their skills of mental estimation. Theyshould also be interpreting and generalizing their answers.

What is the largest number you can make by pressing the keys andonce and only once?

Example:

Can you make a larger number?Using algebra, for any four digits a,b,c,d, wherea

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Junior high schoolDifferent Products

Using the activity in the classroom This activity could be introduced to the whole class by asking students to individually make up any

multiplication using only the digits1, 2, 3 and 4. The different multiplication problems and theiranswers can then be compared and students can be set the task of finding the largest product.Students should be encouraged to estimate the answers to the various multiplication problems.

Points for students to discuss Students can explore other sets of four numbers, generalizing the rule to find the largest productusing words or symbols. After generalizing, explain the rule that for any four digitsa,b,c,d, wherea

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Sums and ProductsJunior high school

Objective Calculate with decimals and understand the results.

Select suitable sequences of operations and methods of computation, including trial-and-improve-ment methods, to solve problems involving integers and decimals.

Explanation of the activity Choose two numbers whose sum is10.Find out what the product of those two numbers would be.Find the products of other pairs of numbers whose sum is10.Find out which number pair gives the largest possible product.

This activity helps to reinforce students understanding of the mathematical terms sum and prod-

uct and develops trial-and-improvement methods.

Using the calculator Calculator functions used: Addition, multiplication, subtraction, parentheses

Try to find the largest product of any two numbers whose sum is10.

Example:

2 8

2 8

You can also calculate this as 2 x (10 - 2) =16.

2 10 2

What two numbers give the largest product?Try multiplying various combinations of numbers whose sum is10.

Ans: 5 x 5 = 25


2 + 8 =DEG

2 X 8 =DEG

2 X( 10 - 2 ) =DEG

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Junior high schoolSums and Products

Using the activity in the classroom This activity could be introduced orally.

The largest product is 25, given by 5 x 5. Some students may need to be encouraged to considerdecimal numbers to verify that the largest product is 25. More able students should be encouragedto try and prove that this is the largest product.One method of using the calculator is to enter the product as two numbers that can be edited.Some students may prefer to enter the product as an expression such as 2 x (10 - 2), which can beedited.

Points for students to discuss Students could be encouraged to devise similar problems to give to each other involving numberswith different sums.

Fur ther Ideas

Investigate products of 3, 4, 5... numbers which have the same sum. This could be exploredgraphically.(Generally, for two numbers whose sum isn, the largest product is given by (n/2)2, for threenumbers whose sum is n, the largest product is given by (n/3)3... The nearest integer to (n/e)wheree= 2.718 is the number of numbers which will give the maximum product.)

The problem of finding two numbers whose product is a given total can be turned into agame where students score points according to the number of trials they perform toidentify the solution. For example: The sum of two numbers is10 and their product is19.71. What are the two numbers?

Ans: The two numbers whose product is19.71 are 7.3 and 2.7.

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Target 100Junior high school

Objective Understand and use the concept of place value in whole numbers and decimals, relating this to

computation.Calculate with decimals and understand the results; e.g. multiplying by numbers between 0 and1.Mentally estimate and approximate solutions to numerical calculations.

Explanation of the activity A game for two players. Player 1 enters any 2-digit number into the calculator. Player 2 then multiplies this by another number so that the answer is as close as possible to100. Players score points according to how close they are to100:

within10 =1 point

within 5 = 2 pointswithin1 = 5 pointsexactly100 =10 points

Player 2 then enters a number and the game continues. The first player to score 20 points wins.

While working on this activity, students will be extending their understanding of decimals andimproving their estimation skills.

Using the calculator Calculator functions used: Multiplication

Example:

Player 1 enters 28.

28

Player 2 multiplies this by 3.5.

3.5

Player 2 scores two points.

The game continues until one player reaches 20 points.


DEG

2 8 X 3 . 5 =DEG

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Junior high schoolTarget 100

Using the activity in the classroom This activity could be given to students with little introduction from the teacher. Alternatively, the

game could initially be played between the teacher and a large group of students. It is importantthat students are encouraged to think carefully about the numbers they choose and that theteacher focuses on the students mental skills. Most benefit is obtained from the activity whenstudents are playing together in small teams, discussing their choices of a number to multiply by.

Points for students to discuss At the end of the activity, students strategies should be discussed and compared.

Further IdeasPlay the game with different target numbers. For example, students could multiply or divide

a random number to reach a target of1.

The first player multiplies a random number to aim for a target of100. The second playerthen multiplies this answer to try and get even closer to100. The player who gets thecalculator to display a number between 99 and101 wins.

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Adding Fractions

Objective Understand and use fractions.

Calculate with fractions and understand the results.

Explanation of the activity Using the calculator, find the sum of two given fractions each having1 in the numerator.Look for patterns to help understand how to add the fractions without using the calculator.This activity suggests an approach to teaching addition of common fractions.

Using the calculator Calculator functions used: Addition, fractional calculation

Example:

Using fractional calculation, find the sum of and .

1 2 1 3

+ = on the calculator display means .

Convert to decimal notation.

0.83333 on the calculator display means .

Find the sums of other common fractions.

15

17

Junior high school/Elementary school

(upper grades)


1 5 +1 7 =DEG

1 5 +1 7 =DEG

1 2 + 1 3 =DEG

1 2 + 1 3 =DEG

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Adding Fractions

Using the activity in the classroom This activity should be presented after studying equivalence of common fractions.

The activity is best introduced orally. Ensure that the students know how to add two commonfractions on the calculator. Ask them to add1/2 and1/3 and record the answer (5/6). Ask thestudents if they can see any connection between the answer and the original two fractions. Stu-dents may note that 2 + 3 = 5 and 2 x 3 = 6. Allow students to explore other unit fractions andencourage them to generalize. Students should be asked to try and explain what is happening. Itshould be noted that the pattern may appear to break down when fractions with a commondenominator are added.

Points for students to discuss

Students can then explore what happens when other common fractions are added. For somestudents, it may be appropriate to begin by considering a pair of fractions that includes one unitfraction.

It is important that students are encouraged to understand what is happening, and that referenceis made to equivalent fractions.

Fur ther Ideas

Investigate subtracting, multiplying or dividing common fractions.

The Babylonians mostly used fractions which had1 as the numerator. For example, 5/6

could be written as1/2 +1/3. Investigate Babylonian fractions.


(upper grades)

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Halfway Between

Objective Understand and use fractions.

Calculate with fractions and understand the results.

Explanation of the activity Use the calculator to find the fraction that is exactly halfway between two other fractions.Look for patterns to help understand how to find the answer without using the calculator.

This activity reinforces addition of common fractions and considers the result of dividingcommon fractions by integers. By working on the activity, students should also develop anincreasing feel for the relative sizes of fractions.

Using the calculator Calculator functions used: Addition, division, multiplication, fraction, calculation

Example A:

Find the fraction that is halfway between and .

Using fractional calculation, obtain the sum of and .

1 2 1 3

Half of this fraction is the number you are looking for, sodivide this fraction by 2.

2

Or after , multiply by .

1 2

Example B:

Find the fraction that is halfway between and .

Using fractional calculation, obtain the sum of and .

1 3 1 4


(upper grades)


1 3 + 1 4 =DEG

A N S X 1 2 =DEG

A NS2 =DEG

1 2 + 1 3 =DEG

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Halfway Between

Using the activity in the classroom This activity could follow the study of addition of common fractions.The activity is best introduced orally. Ask the students to give different fractions that lie between 5/12 and 2/3. One possibility is to arrange these on a fraction line. It is important that students are

challenged to justify their answers and, in some cases, it may be appropriate to consider decimalequivalents. The students should then be asked to identify the common fraction that is halfwaybetween 5/12 and 8/12, justifying their answer.

Points for students to discuss Furthering the activity, students can be asked to give fractions that lie between1/2 and1/3 andidentify the common fraction that is halfway between them. At this stage it may be necessary todiscuss methods for finding a number that is halfway between two numbers. Students can then usetheir calculators to identify fractions that are halfway between other unit fractions. This can beextended to non-unit fractions. It is important that students are encouraged to understand what ishappening.

Fur ther Ideas

Find fractions that lie1/3 of the way between two fractions, or 1/4 of the way between twofractions, etc.


(upper grades)

Half of this fraction is the number you are looking for,so divide this fraction by 2.

2

Or after , multiply by .

1 2

Continue the activity using other common fractions.

A N S X 1 2 =DEG

A NS2 =DEG

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Near IntegersJunior high school

Objective Understand and use the concept of place value in decimals.

Understand and use decimals and fractions while comprehending the interrelationship betweenthem.Use some common properties of numbers, including multiples.Give solutions in the context of the problem, selecting the appropriate degree of accuracy andinterpreting the display on a calculator.

Explanation of the activity Use the calculator to find fractions that are near integers in decimal form.

While working on this activity, students will be developing their understanding of decimals,

particularly their relationship with fractions.

Using the calculator Calculator functions used: Multiplication, division

The 35th multiple of 0.314 is10.99.35 x 0.314 =10.99

35 0.314

10.99 is a near integer; it is nearly11.

Using fractional calculation, input .

11 35

Convert to decimal notation.

Using division, divide11 by 35.

The fraction has a decimal value close to 0.314.

11 35


113 5 =DEG

11 3 5 =

DEG

3 5 X0 . 3 14 =DEG

11 3 5 =DEG

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Junior high schoolNear Integers

Using the activity in the classroom This activity is probably best introduced orally. Students could use the sequence function of the

calculator to generate the multiples of some integers, and could then begin to investigate themultiples of some decimals.

Points for students to discuss The teacher could ask the students to generate the multiples of 0.314, challenging them to find amultiple that is nearly an integer. Students can then begin to investigate the situation further.

Fur ther Ideas

Use this idea to investigate different approximations for . For example, 22/7 = 3.142857,whereas 3.141593. However,179/57 = 3.140351.

Investigate approximations for , or , etc.

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Reshaping CuboidsJunior high school

Objective Reinforce students understanding of the equivalence of shapes in various alignments and how this

relates to multiplication within a practical context.Develop mental skills involving factors, divisors, and systematic thinking.

Explanation of the activity 12 cubes, each with a volume of1 cm3, may be placed together to create any of four cuboids, eachhaving a volume of12 cm3.

Find the equivalent equations for each of the cuboids; for example,1 x1 x12 =12,2 x 2 x 3 =12, etc.

Using the calculator Calculator functions used: Multiplication, Multi-line Playback

Introduce students to the calculators Multi-line playback feature, which will be useful todisplay sets of solutions for each volume number.

1 1 12

2 2 3

1 3 4 etc.


1X3 X4 =DEG

2 X 2 X 3 =DEG

1X1X12 =

DEG

1X3 X4 =DEG

2 X 2 X 3 =DEG

1X1X12 =DEG

Each press of a key takes you one calculation step forward or backward.

Display the first calcualtion

with

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Junior high schoolReshaping Cuboids

Using the activity in the classroom Students may benefit from the use of actual blocks that can be stacked to form the different cubiccombinations. An OHP calculator could also be used to collect solutions from the entire class.

Points for students to discuss The number of divisors for a number expressed aspa x qb x rc (wherep,q, and rare all prime) is(a+1) (b+1) (c+1). For example, 360 = 23 x 32 x 51. Here,a= 3,b= 2, and c=1, so the numberof divisors is given by the expression (3 +1) (2 +1) (1 +1) = 24. Therefore, 360 has 24 divisors.

Fur ther Ideas

Use trial and improvement to find the side of a cube having a volume of180 cm3.

Move into four (or more) dimensions as a means of finding the factors of a number.For example, 6006 = 77 x 78 = (7 x11) x (6 x13) = 2 x 3 x 7 x11 x13. All stages canbe displayed using the replay function.

1 x1 x12 =12 1 x 2 x 6 =12

2 x 2 x 3 =12 1 x 3 x 4 =12

Find the five calculations that representcuboids that each have a volume of 30 cm3.

e.g. 1 1 30 etc.

In a similar way, find the twelve calculations forcuboids each having a volume of 96 cm3.

How many similar calculations must there befor 180 cm3?Which of these cuboids is nearest to looking

like a cube?

For volumes between150 cm3 and 200 cm3,which particular ones can be represented by atleast 16 cuboids each? Which volumes havethe smallest number of cuboids each?

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Function TablesJunior high school

Objective Understand and use calculator functions.

Understand and apply functional relationships.Enable speedy plotting of graphs.

Explanation of the activity Use the calculator to calculate theyvalues for a given function using a set range of values for x.Record the values on a table and use them to plot a graph.

Using the calculator Calculator functions used: Multiplication, editing, Multi-line Playback

Enter theyvalues for the functiony= 4x- 5 using the values from -5 to +5 for x.Use the calculators playback function to calculate the functions efficiently. After calcu-lating the values, use them to plot the graph ofy= 4x- 5.

4 5 5

4


Each press of a key takes youone calculation step forward or backward.

x -5 -4 -3 -2 -1 ... 5

y -25 -21 -17 -13 -9 ... 15

In the same way, find the values for

x=-3, -2,1, 0,...5.

4 X ( - 5 ) - 5 =DEG

4 X ( - 4 ) - 5 =DEG

...

...

...

...

4 X ( 5 ) - 5 =DEG

4 X ( 4 ) - 5 =DEG

4 X ( - 4 ) - 5 =DEG

4 X ( - 5 ) - 5 =DEG

4 X ( - 4 ) - 5 =DEG

4 X ( - 5 ) - 5 =

DEG

Display the first calcualtion with

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Junior high schoolFunction Tables

Using the activity in the classroom This activity should be introduced after practicing substitution.

Start the activity as a whole class so the students can gain confidence in using the calculator andsee the advantages of calculating first and then recording the results to speed up the process ofmaking the graph table. The students can calculate theyvalues for the second equation themselvesand quickly continue with other suggested equations using multi-line playback to go directly fromthe (x,y) values to the graph without needing to record the result in a table. This enables thefamilies of graphs to be compared rapidly. Try extending the activity by using graphs with differentgradients to establish the parallel nature of the graphs, and then try keeping the intercept constantand varying the gradient.

Points for students to discuss The idea of using the playback function as a rapid way to calculate function values can be applied toa wide range of equations including polynomials, trigonometric functions, etc. Students can docalculations in one sequence and then use the playback function to go back through the answersand record or plot them all at once.

Fur ther IdeasInvestigations on graphs can be done more quickly if the playback function is used so eachfunction does not have to be retyped at every entry. Demonstrate this by using the followingsuggestions:

Solve a quadratic function such asax2 +bx+c= 0 for varying values ofa,b, and c.

Use the calculator to generate values of a trigonometric function and enter the resultsdirectly onto a graph using the playback function.

Return to the calculation for x= 5 and redo the calculation for the equationy= 4x- 3.Add another line to the table and calculate the values ofyfor the new equation.Plot the second graph with the first on the same axis.

What do you notice about the graphs and the numbers in the table?

What do you think will happen if you try another similar equation such asy= 4x- 1,y= 4x+1, or y= 4x+ 4?

Can you explain the number pattern and the picture you have produced?

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Write down some 4-digit palindromes and use the calculator to divide each of them by11.

Example:

2332 11

5665 11 etc.

Do you notice a pattern?

The use of the playback function will speed up the calculations and enable students tocompare results to look for a pattern.

Make up some more 4-digit palindromes and divide each of them by11.Have students compare their results until they notice a pattern.

PalindromesJunior high school

Objective Understand and use the concept of place value in whole numbers.

Explore a variety of situations that lead to the expression of relationships.Construct and interpret formulas and expressions.Manipulate algebraic expressions.

Explanation of the activity A word that reads the same forwards and backwards, such as mom and level, is called a palin-drome. A palindromic number is exactly the same; the number has the same value whichever wayyou write the digits. For Example, 212, 34543 and10001.

Using the calculator

Calculator functions used: Division


5 6 6 5 K=DEG

2 3 3 2 11=DEG

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Junior high schoolPalindromes

Using the activity in the classroom This is an activity that can first be given to students to work on and the results later discussed as a

group. The object is to discover a pattern in the resultsthe pattern being that the answer willalways be a whole number.

Points for students to discuss The pattern may be too obvious for the students to mention, so it may be necessary to give themthe following hint:Try writing the number in its long form; for example,

2332 = 2 x1000 + 3 x100 + 3 x10 + 2

Fur ther IdeasTry to explain the problem using algebra. You could start off by giving the class just the firstline of the calculation below and let them work on the rest individually or in groups.

a b b a =ax1000 +bx100 +bx10 +a=ax1000 +a+bx100 +bx 10=a(1000 +1) +b(100 +10)=ax1001 +bx110=a91 x11 +bx10 x11=11 (91a+10b)

Try the activity using 6-digit palindromes.Have the class prove that not all 5-digit palindromes are exactly divisible by11.

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Trial and ImprovementJunior high school

Objective Mentally estimate and approximate solutions to numerical calculations.

Understand and use the concept of place value in decimals and relate it to computation.

Explanation of the activity Use trial and improvement to find the length of the side of a cube-shaped box that can hold100cm3 of ice cream.The two mental calculations 4 x 4 x 4 = 64 and 5 x 5 x 5 =125 should suggest a possible startingcalculation such as 4.5 x 4.5 x 4.5 = 91, which can be shortened to 4.53= 91.This activity gives students the opportunity to enhance their understanding of decimals and im-prove their skills in estimation.

Using the calculator Calculator functions used: Multiplication, FSE, TAB

Set the calculator to fixed point notation with a TABvalue of 0.(Doing this will display answers to the nearest whole number.)

Adjust the TAB setting to 1 and then continue to improvethe accuracy of the answer

1

4 4 4

5 5 5

4.9 4.9 4.9

4.7 4.7 4.7

4.6 4.6 4.6

From this we can see the answer lies between 4.6 and

4.7. Continue to search for the answer repeating thisoperation.


DEG

F I X DEG

4 . 642X4 . 642XF I X DEG

4 . 6 X 4 . 6 X 4 . 6 =F I X DEG

4 . 7 X 4 . 7 X 4 . 7 =F I X DEG

4 . 9 X 4 . 9 X 4 . 9 =F I X DEG

5 X 5 X 5 =F I X DEG

4 X 4 X 4 =F I X DEG

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Junior high schoolTrial and Improvement

Using the activity in the classroom This activity may be given to students with little introduction or, with the use of the OHP unit, thisor a similar task may be introduced to the whole class followed by individual work on one or moreof the extension activities. The use of the multi-line playback function will be of practical benefit intackling questions involving trial and improvement.

Points for students to discuss It will be necessary to familiarize the students with the FSE and TAB keys in order to understand,for example, why 4.6413 and 4.6423 both have the value100 to the nearest unit. In the context ofsimilar problems, students will need to consider what degrees of accuracy are appropriate; in thecase of cubic centimeters of ice cream, possibly only to one decimal place.

Fur ther Ideas

Find the side of a cubical carton whose volume is1/2 liter. It may be necessary to remindstudents of the equivalence of 500 ml (fluid measure) and 500 cm3 (solid measure).

Find the dimensions of a fruit juice carton whose sides are in the proportion1 : 2 : 3 and

whose capacity is1

liter.

Find the Golden Ratio xby trial and improvement of the relation

Guessx(Guess +1) =1

Use the playback function on the calculator to show that

x=1 / (1 +x) and that x= .

All metric paper has the same shape (except golden). If A0 has an area of1 m2 and the longer

side is times bigger than the smaller side, find these dimensions. What are the dimensionsof A4? Have the students confirm their calculations by measuring a sheet.

Switch FSE and TAB to normal display for further operation.

Press FSE until FIX, SCI, or ENG are not shown on the display.

DEG

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Last DigitsJunior high school

Objective Use last digits as a means of checking the output of a calculator.

Practice estimation and observe patterns.Reinforce the concept of prime numbers.

Explanation of the activity Perform a series of multiplication equations keeping the last digit of each of the multipliers con-stant; for example, 3 x 7, 13 x 7, 3 x17, etc.

Using the calculator Calculator functions used: Multiplication

Enter the following equations into the calculator:

3 7

13 7

3 17

Find other last digit combinations that give answers ending in1.

Which of the numbers in the following set can be made from the product of two numbers?(excluding equations using1 multiplied by the number itself)

21, 41, 51, 61, 71, 81, 91,101,111,121,131,141,151,161,171,181,191, 201

Which of the numbers can be made in more than one way?

Make a collection of your calculations so that they can be displayed in order of answer size.

Name the type of numbers that cannot be made.


3X17=DEG

13X7=DEG

3 X 7 =DEG

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Junior high schoolLast Digits

Using the activity in the classroom It is probably best to first introduce the activity as a class to give the students an opportunity to

make estimates before using their calculators. Once the class has shared their initial ideas, they canbe given time to investigate any patterns they discover.

Points for students to discuss After investigating patterns on their own, students should share their discoveries with the rest ofthe class.

Fur ther Ideas

Examine the first 20 prime numbers. Except for the number 2, they all end with an odd lastdigit. Repeat the procedure for the last digits of 3, 7, and 9. Find all the prime numbers

between1 and 201.

Find the last digits to:1. the answers to the multiplication tables.2. the square numbers.3. other number sequences such as the cube numbers and triangle numbers.

Find the two consecutive numbers whose product is 6006.

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34

A Question of InterestJunior high school

Objective Understand, use and calculate with percentages.

Select suitable sequences of operations and methods of computation, including trial-and-improve-ment methods, to solve problems involving integers, decimals and percentages.Give solutions in the context of the problem, selecting an appropriate degree of accuracy, andinterpret the display on a calculator.

Explanation of the activity Use the calculator to find solutions to problems involving interest rates.While working on this activity, students will develop their understanding of percentages within thecontext of compound interest situations.

Using the calculator Calculator functions used: %calculation, multiplication


If you invest money at a certain level of interest, by how much will your money grow?

Example:

$100 is invested at 0.1%annual interest.

Using multiplication: Multiply the principal $100 by1.001.

100 1.001

For the total after two years, multiply the previous answeragain by1.001.

1.001

After three years...

After four years...

After 10 years, you have approximately $101.

10 0 X1.0 0 1=DEG

ANSX1. 0 0 1=DEG

ANSX1. 0 0 1=DEG

ANSX1. 0 0 1=DEG

ANSX1. 0 0 1=DEG

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Junior high schoolA Question of Interest

Using the activity in the classroom This activity is probably best introduced orally. After a discussion about investments and interestrates, the teacher can use the sequence function of the calculator to generate sequences showinghow an initial capital sum grows for a fixed interest rate. Students can be asked to find the annual

interest rate that ensures their money is doubled in10 years.Students can then investigate the annual interest rates that would double their money for differentnumbers of years. These interest rates could be plotted on a graph.

5 years 14.9%10 years 7.2%15 years 4.7%20 years 3.6%25 years 2.8%

Points for students to discuss It may be useful to show students how to generate sequences on the calculator.

Fur ther Ideas

Investigate interest rates that would triple an investment, or...From1970 to1980 prices tripled. What was the average rate of inflation?

Using the %calculation key: After one year, you should have 0.1%of your$100.

100 0.1

You now have $100.10.

After two years, you have 0.1%more. 0.1

After three years...

After 10 years...

You have approximately $101. ANS+0 .1%DEG

ANS+0 .1%DEG

ANS+0 .1%DEG

10 0 +0 .1%DEG

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Junior high schoolA Question of Interest

For High school Students

How much will your investment be worth innyears?Lets make an equation.

The original amount of money invested, called the principal, multiplies each year by theamount x.Lets use this equation to see how much money we have after100 years.

The original amount, or principal, is $100; so a=100.

The number of years is100; so n=100.The interest is 0.1%; so x=1.001.

100 1.001 100

You have $110.50 after 100 years.

How many years would it take for the money to double?Lets make an equation.The money invested multiplies each year by the amountx.After nyears the money doubles, so...

axn= 2aDivide both sides by a xn = 2Calculates the log of both sides logxn = log2

nlogx= log2

n = log2/logx

Ifais the money deposited, the savings would double.

2 1.001

It takes approximately 694 years for your money to double.


10 0 X1.0 0 1 10DEG

log2 log1.0 0DEG

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37

Getting EvenElementary school

(upper grade)

Objective Use some common properties of numbers.

Explore a variety of situations that lead to the expression of relationships.

Explanation of the activity A game of chance to compare the relationship between odd and even numbers.By working on this activity, students will reinforce their understanding of odd and evennumbers. More able students could develop their skills in using algebra to prove generaliza-tions.

Using the calculator Calculator functions used: Addition

A game for two players

The first player enters any number into his/her calculatorwithout showing it to the other player.For example, 298.

The second player then enters a number into his/hercalculator without showing it to the other player.For example, 55.

The players then show each other their numbers and addthem. If the answer is even the first player scores1 point; ifthe answer is odd, the second player scores1 point.

298 55

The first player to score10 points is the winner.


2 9 8 + 5 5 =DEG

DEG

DEG

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38

Elementary school(upper grade)

Getting Even

Using the activity in the classroom The game is best played between pairs or small groups of students. It could be introduced by the

teacher playing the game against some students.While playing the game, students should be encouraged to reflect on whether the game is fair, andalso try and think about the reasons for their conjectures.

Odd + Even = OddEven + Odd = OddOdd + Odd = EvenEven + Even = Even


More able students could try to formally prove their conjectures.The idea can be extended by students thinking about the conditions for obtaining even or oddanswers when three numbers are added, or four numbers, or...

Fur ther Ideas

More able students could try to formally prove their conjectures.The idea can be extended by students thinking about the conditions for obtaining evenor odd answers when three numbers are added, or four numbers, or...

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39

Generating SequencesJunior high school

Objective Appreciate the use of letters to represent variables.

Explore number patterns arising from a variety of situations.Interpret, generalize and use simple relationships, and generate rules for number sequences.

Explanation of the activity Express simple functions initially in words and then symbolically.Initially this will be in words, using ANS to represent the variable, but this should lead to using theletter a.Through discussion, it is possible to develop understanding of algebraic equivalence; for example a+ a is equivalent to 2 x a, which is often written as 2a. This can lead to simplifying algebraicexpressions.

While working on this activity, students will be developing skills of generalization and refining theirmethods of expressing mathematical rules.

Using the calculator Calculator functions used: Addition, subtraction, multiplication, division,

last answer memory

Lets look at a sequence, find a rule to generalize it, and check it with our calculators.

Example:3, 5, 7, 9, 11

The above sequence can be described as an ANS + 2 sequence. This can be confirmedby adding 2 to 3, then 2 to this sum, and so on.

Add 3 and 2.

3 2

Compare the result of your calculation with each number in the sequence.

Add 2 to this sum.

2

Again, add 2 to this sum.

( 2 )

And so on.

We were thus able to confirm our ANS + 2 rule.

Try making rules to describe other number sequences.


3+2=DEG

ANS+2=DEG

ANS+2=DEG

ANS+2=DEG

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40

Junior high schoolGenerating Sequences

Using the activity in the classroom This activity could be introduced orally. Read the terms of a sequence one at a time, asking students

to identify the rule you are using. Repeat for other sequences and discuss the different rules.Below are some possible solutions. It is important to emphasize that there may be differentsolutions.

ANS + 4ANS - 52 ANSANS 3ANS x 3 - 1ANS2 + 1

Points for students to discuss It may be useful to show students how to generate sequences on the calculator.

Further IdeasFind different sequences that begin 1, 2, 5...Is it possible to generate the multiples of any number, square number, triangle number?

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41

Train JourneysJunior high school

Objective Develop and understand the relationship between units.

Understand and use compound measures including speed.Undertake purposeful inquiries based on data analysis.

Explanation of the activity This activity provides students with the opportunity to handle real data and work on a wide rangeof activities involving time, distance and speed.

Using the calculator Calculator functions used: Degrees/minutes/seconds key (DMS),

time calculation (substraction, division)

The information on the table below shows the departure and arrival times of twotrains, together with the distance of each station from Norwich.How long does it take to complete the various stages of the journey?

NorwichDiss 17.75DissStowmarket 13.5

StowmarketIpswich 10.25IpswichColchester 15.25ColchesterChelmsford 19.5ChelmsfordLondon 26

Example:Try calculating the average speed of a train traveling from Norwich to London. Thetrain leaves Norwich at 11:30 and arrives in London at 13:22. Therefore, the time forthis journey is calculated by subtracting 11:30 from 13:22.

Input13:22.13 22

Subtract 11:30

11 30

We can see the journey takes one hour and 52 minutes.How many hours is this?Lets find out.

One hour and 52 minutes is 1.8666 hours.


132200.00DEG

DEG

132200.00DEG

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42

Junior high schoolTrain Journeys

Using the activity in the classroom This activity could be used with small groups, with each group looking at different aspects of thesituation. For example, mean distances, times and speeds and relationships between distance andtime. Students could investigate the average speeds for various stages of the journey, consideringreasons for the differences in average speed.

Points for students to discuss Students could use the degrees/minutes/seconds key (DMS) to enter and compute various times.The actual purpose of the key should be discussed with students, however.

Further Ideas Compare other train lines and other forms of transport.

What factors affect the speed of a train?

Investigate the statement:Longer journeys are quicker.

Lets find the average speed at which the train travels between Norwich and London.

Speed = distance time115 miles 1:52 is calculated as:

115 1 52

The train travels at approximately 61.6 mph.

How long does it take to complete the various stages of the journeys?

Why is the train that starts at Colchester described as a slow train?

What conjectures can you make about the different stages of the journeys?

11515200.DEG

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43

Simulated DiceJunior high school

Objective Use computers as a source of large samples and as a means to simulate events.

Understand and use relative frequency as an estimate of probability, and judge when sufficient trialshave been carried out.Recognize situations where probabilities can be based on equally likely outcomes.

Explanation of the activity Use the calculators to simulate rolling dice.

While working on this activity, students will be developing a feel for probability. In particu-lar, the relationship between estimating probabilities from experimental evidence and calcu-lating probabilities based on equally likely outcomes.

Using the calculator Calculator functions used: Random numbers, addition, fixed number calculation

Lets use our calculators to simulate rolling dice.

Example:

To simulate rolling dice, we want the numbers from1

to 6,so the formula will be1 + RANDOM(5).

However, this will also generate decimals. So, change tothe FIX mode by pressing TAB and 0 until appears FIX ona screen. Now only integers from 1 to 6 will be generated.

Set TAB to zero so that values to the right of the decimalpoint will not be displayed. (Values to the right of the

decimal point are rounded to the nearest whole number.)0

To generate random numbers from 1 to 6, calculate theformula 1 + RANDOM(5).

Try this numerous times.

1 5

Lets use the calculator to simulate the results obtained byadding the scores on two 16 dice.


*1 It is the nature of random numbersthat the value may not be the same foreach time.

......

FIX DEG

FIX DEG

1+RANDOM(5)=

*1FIX DEG

1+RANDOM(5)=*1FIX DEG

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44

High schoolSimulated Dice

Using the activity in the classroom This activity could form the basis of a whole class investigation. After some initial work on prob-ability, students could be introduced to the idea of using the calculator to simulate dice.The question can then be raised about the reliability of the simulation. The calculator can be usedto simulate a die and the results can be collected, presented and analyzed. If the whole class works

on the problem, a large number of trials can be carried out quite quickly. The results can be com-pared to expected frequencies based on equally likely outcomes.Other ways of using the calculator to simulate dice can also be explored. For example, 0.5 +RANDOM (6). Students could then explore the reliability of other dice simulations, such as addingor multiplying the scores on two dice.

Points for students to discuss It is probably best to explain orally to students how to use the calculator to simulate dice beforestarting the activity.

Further Ideas Compare the calculators simulation of dice to a computers simulation of dice.

After finishing this example, it is advised to return the display mode back to normal.

Lets create a simulation that throws these dice and finds their sum.[1+RANDOM (5)] + [1+RANDOM (5)]How good a simulation is this? Investigate other dice dimulations

DEG

(Repeat until the FIX, SCI,ENG symbols are disappeared)

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45

Mean Dice ScoresJunior high school

Objective Use computers as a source of large samples and as a means to simulate events.

Calculate, estimate and use appropriate measures of central tendency with discrete data.

Explanation of the activity Use the calculator to find out the mean score of consecutive rolls of a single die.

While working on this activity, students understanding of the mean are reinforced. Theactivity also develops students understanding of the concept of a limit, within the context ofprobability, and emphasizes the importance of experimentation and simulation.

Using the calculator

Calculator functions used: Random numbers, addition

Lets use our calculators to find out the mean score if we roll a die 10 times.

The mean is calculated by dividing the total score by the number of rolls of the die.Calculate (4 + 4 + 1 + 5 + 2 + 3 + 6 + 5 + 4 + 6 ) 10.

( 4 4 1 ... 6 ) 10

Lets see what happens to the mean score if we rollthe die 20 times, 50 times, or 100 times.What do you think will happen to the mean score as we continue to roll the dice?

Example 2:Enter the data using statistics mode.

*You may not get the same resultsbecause random numbers are used.

Example 1:


1Stat x

DEG

n=DEG

n=DEG

n=DEG

X=DEG

...

...

...

4

4

6

*x

(4+4+1+5+2+3DEG

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46

High schoolMean Dice Scores

Using the activity in the classroom This activity could be introduced to a group using a real die. After some practical work, the need

to roll the die many times suggests that the data may best be generated by computer simulation.The mean score for one die should approach a limit of 3.5It is important that students are encouraged to try and discover and prove any rules.

Points for students to discuss It is probably best to explain orally to students how to use the calculator to simulate dice beforestarting the activity. If the calculator is set to STAT mode, it is possible to store the rolls of the dieand use the calculator to find the mean score. It is important to note that the calculator should beset back to floating decimal place mode when calculating the mean score. Alternatively, studentscould work in pairs using one calculator to simulate the die and another calculator to store the

data.

Further Ideas Investigate the mean scores obtained by rolling a 18 die, or a 112 die, etc.

Investigate the mean scores obtained by adding the scores on two dice, or adding thescores on three dice. For example, adding two dice may be simulated on the calculator by:

(1 + RANDOM(5)) + (1 + RANDOM(5)).

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Again & AgainHigh school

Objective Explore number patterns arising from a variety of situations.

Interpret, generalize and use simple relationships, and generate rules for number sequences.

Explanation of the activity Express simple functions initially in words and then symbolically.This activity develops students understanding of convergent sequences and the concept of a limit.For more able students, attempting to prove the results will give practice in algebraic manipulation.This can also be used to introduce an iterative method for solving quadratic equations.


last answer memory

Lets see what happens when we repeat the same equationover and over.

Think of a number.Divide it by 2 and add 1.

Example:Try the number 6.

6 2 1

Divide the answer by 2 and add 1.

2 1

Keep on repeating this.

( 2 1 )

What eventually happened?Did the answer come closer and closer to a certain integer?

Try investigating different starting values and different rules to use for therepeat calculation.


ANS2+1=DEG

62+1=DEG

ANS2+1=DEG

ANS2+1=DEG

......

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49

FibonacciHigh school

Objective Appreciate the use of letters to represent variables.

Explore number patterns arising from a variety of situations.Interpret, generalize and use simple relationships, and generate rules for number sequences.Express simple functions symbolically.

Explanation of the activity Use the calculator to generate the Fibonacci sequence.

While working on this activity, students will be developing skills of generalization and refin-ing their methods of expressing mathematical rules.


last answer memory, Multi-line Playback

The Italian mathematician Fibonacci discovered this sequence:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

To generate a similar sequence, begin with two numbers and add them to get the next.Continue adding the last two numbers to get the next term.

Lets try making a regular series similar to the Fibonacci series above. First, select twosuitable integers.


Example:Lets try making our own similar sequence,starting with 2 and 6.

[2, 6, ?]

Adding the two starting numbers yields the third valuein the series.

2 6

[2, 6, 8, ?]

The next value is the sum of the two terms preceding it.Thus

6 8

[2, 6, 8, 14, ?]

Similarly, find subsequent terms in the series by addingthe preceding term and the one before it.

8 14

2+6=DEG

6+8DEG

8+14DEG

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50

High schoolFibonacci

Using the activity in the classroom Introduce the sequence to the students and give them the opportunity to try to generate it ontheir calculators. After some discussion the students can be shown how to use the temporarymemories to generate the sequence. One method of introducing this is to stick two envelopesonto the board, one labeled X and the other labeled Y, in which can be placed pieces of papermarked with the appropriate numbers.The ratio of successive terms of any such sequence converges to the golden ratio,

(1 + ) 2 1.61803399.

Points for students to discuss Explain orally to students how to generate the Fibonacci sequence on the calculator.

Further IdeasAsk students for two starting values and then generate the first ten terms of the sequence.

Predict the sum of these of ten numbers, and then ask the students to confirm your answer.(The trick is that the sum of these numbers will always be the sixth number multiplied by11.)

[2, 6, 8, 14, 22, ?]

Find additional members of the series by repeating the above operation.

[2, 6, 8, 14, 22, 36, 58]

This yields the series:

Recall the first formula

Try generating similar series using a varietyof other numbers.

For each of your sequences, once you havegenerated a large number of terms, dividethe last term by the preceding term andnote the answer.

......

2+6=DEG

6+8DEG

8+14DEG

14+22DEG

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Factorizing QuadraticsHigh school

Objective Evaluate formulas and expressions.

Manipulate algebraic expressions.

Explanation of the activity Confirm the equationx2 + 5x+6 = (x+ 2) (x+ 3) is always true by plugging in values forx.

While working on this activity, students will be learning about the factorization of quadraticexpressions. Some students may also study the graphs of factorized quadratic functions.

Using the calculator Calculator functions used: Addition, subtraction, multiplication, division, squaring,

memory calculation, Multi-line Playback

Example:Confirm the equationx2 + 5x+6 = (x+ 2) (x+ 3) is always true.

(1) Store any number,x, into your calculator.

(2) Use your calculator to evaluate x2

+ 5x + 6(3) Now use your calculator to evaluate (x + 2) (x + 3)

Store 14.

14

Calculatex2 + 5x+ 6.

5 6

Calculate (x+ 2)(x+ 3).

2 3

Press the following buttons and then start operation.(Variablexprinted in blue under one of keys)

(X+2)(X+3)=DEG

X2+5X+6=DEG

14 X DEG

Repeat for other values ofxuntil you are surethe equation always holds true.

(X+2)(X+3)=DEG

X2+5X+6=DEG

Check with Multi-Line Playback

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High schoolFactorizing Quadratics

Using the activity in the classroom It is assumed that students will have done some work on quadratic functions prior to this activity.For example, students may have been attempting to solve a range of quadratic equations given in avariety of forms. Some of these equations could possibly be solved via inverse relationships, (e.g.x2 - 4 = 15), while some may be already given in factorized form and so could be solved directly.Others may have to be solved by trial and improvement. This could lead to students concludingthat factorizing is a useful technique in solving quadratic equations.The activity could then be introduced as an investigation into factorizing quadratic expressions.The teacher could do an example on the board, and then other expressions could be givento the students to factorize, using trial-and-improvement methods on their calculator.

It is important that students are encouraged to reflect on the factorized expressions and lookfor strategies to help them factorize quadratics. These strategies should be collated togetherand discussed.

Points for students to discuss It may be useful to go over with some students how to enter and evaluate algebraic expressionson the calculator.

Further IdeasThe idea can be adapted to focus on the equivalence of other algebraic expressions, includingfactorizing linear expressions such as 4x+ 6.

Use the calculator in the same way to find similar equations for the expressions below.

x2 + 7x+ 10x2 + 7x+ 12x

2 + 6x+ 8x2 + 10x+ 16x2 +x- 6x

2 + 5x+ 14

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TriplesHigh school

Objective Understand and use Pythagoras theorem.

Use coordinate systems to specify location.

Explanation of the activity Working on this activity will reinforce students understanding of Pythagoras theorem, and developtheir knowledge of Pythagorean triples. It will also develop their appreciation of different ways ofspecifying position, including rectangular Cartesian coordinates and polar coordinates.

Using the calculator Calculator functions used: Coordinate conversion

Try to find the length of the hypotenuse (the sideopposite the right angle) of a right triangle whoseother two sides are 3 cm and 4 cm long.

Input the lengths of the two shorter sides.

3 4

Find the length of the hypotenuse.

We find that the length of the hypotenuse is 5 cm.

Extra:Try to find the angle of the corner between thehypotenuse and the base.

We find that this angle is 53.13010235.

Lets find the hypotenuses of right triangles havingvarious bases and heights.


(Repeat until the DEG symbol appears)

3,DEG

r=DEG

=DEG

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High schoolTriples

Using the activity in the classroom This activity assumes that students have done some previous work on Pythagoras theorem. After

a short introduction, the students can be set the task of trying to find right-angled triangles wherethe sides are all integers. The students can be shown how to use the rectangular-polar coordinateconversion function of the calculator to find the hypotenuse given the two shorter sides. It isimportant that it is explained to students that the calculator is converting a pair of rectangularCartesian coordinates to polar coordinates, which comprise a distance and an angle.Students should be encouraged to identify the different families of right-angles triangles, (e.g. 3, 4,5 and 6, 8, 10 etc.,) and concepts of similarity should be discussed. It is important that students generalizations are collated and discussed.


To generate a Pythagorean triple, square any odd number and split the answer into two consecu-tive integers. The original number and the two consecutive integers will form a Pythagorean triple.For example: 52 = 25 = 12 + 13. Alternatively, square any even number, divide it by two, and thensplit it into two close integers. The original number and the two close integers will form aPythagorean triple. For example: 62 = 36; 36 2 = 18 = 8 + 10.

Further IdeasWhen are the areas and perimeters of right-angled triangles numerically equal? Investigate.

A consideration of the angles obtained by converting rectangular Cartesian coordinates topolar coordinates could form the basis of an introduction to trigonometry.

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